| fullrecord |
[{"key": "dc.contributor.author", "value": "Gr\u00f6n, Jenni", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2019-06-10T07:11:11Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2019-06-10T07:11:11Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2019", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/64447", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tutkia vakioleveit\u00e4 joukkoja ja niihin liittyvi\u00e4\ntuloksia. Vakioleve\u00e4t joukot ovat joukkoja, joiden leveys jokaiseen suuntaan on yht\u00e4\nsuuri. Tasossa t\u00e4m\u00e4 tarkoittaa sit\u00e4, ett\u00e4 rajattaessa joukko kahden yhdensuuntaisen\nsuoran v\u00e4liin siten, ett\u00e4 suorat juuri koskettavat joukkoa, suorien et\u00e4isyys on vakio\nriippumatta siit\u00e4, miss\u00e4 kohtaa suorat sivuavat joukkoa. Yksinkertaisin esimerkki\nvakioleve\u00e4st\u00e4 joukosta on kiekko, mutta joukkoja on muitakin. Er\u00e4s tunnetuimmista\nvakioleveist\u00e4 joukoista on Franz Reuleaux\u2019lta nimens\u00e4 saanut Reuleaux\u2019n kolmio, joka\nmuodostetaan tasasivuisesta kolmiosta yhdist\u00e4m\u00e4ll\u00e4 viereiset kolmion k\u00e4rjet ympyr\u00e4n\nkaarella.\nTy\u00f6ss\u00e4 l\u00e4hdet\u00e4\u00e4n liikkeelle konveksin joukon m\u00e4\u00e4ritelm\u00e4st\u00e4 sek\u00e4 tutustutaan konveksien\njoukkojen geometriaan yleisess\u00e4 avaruudessa. Lis\u00e4ksi tutustutaan hypertason\nja tukitason m\u00e4\u00e4ritelmiin ja todistetaan, ett\u00e4 kompaktin ja konveksin joukon jokaisessa\nreunapisteess\u00e4 on olemassa tukitaso. Seuraavaksi siirryt\u00e4\u00e4n joukon leveyden m\u00e4\u00e4rittelyyn,\nmist\u00e4 p\u00e4\u00e4st\u00e4\u00e4n luontevasti tutkimaan vakioleveit\u00e4 joukkoja. T\u00e4m\u00e4n j\u00e4lkeen\nrajoitutaan tasoon, sill\u00e4 vaikka vakioleveit\u00e4 joukkoja on tutkittu melko paljon, niist\u00e4\ntiedet\u00e4\u00e4n suhteellisen v\u00e4h\u00e4n R2:sta korkeammissa ulottuvuuksissa.\nTy\u00f6n p\u00e4\u00e4tuloksena todistetaan Barbierin lause, jonka mukaan vakioleveiden joukkojen\npiiri lasketaan kertomalla joukon halkaisija piill\u00e4, kuten lasketaan my\u00f6s ympyr\u00e4n\npiirin pituus. Lis\u00e4ksi todistetaan, ett\u00e4 edell\u00e4 mainittu Reuleaux\u2019n kolmio on\nvakioleve\u00e4 ja annetaan muita esimerkkej\u00e4 tason vakioleveist\u00e4 joukoista. Kaksiulotteisten\nvakioleveiden joukkojen pinta-aloille voidaan my\u00f6s antaa rajat. Annetun levyisist\u00e4\nvakioleveist\u00e4 joukoista suurin pinta-ala on kiekolla ja pienin pinta-ala on Reuleaux\u2019n\nkolmiolla. Lopuksi esitell\u00e4\u00e4n hieman kolmiulotteisten vakioleveiden joukkojen eli kappaleiden\nmuodostamistapoja sek\u00e4 esimerkkej\u00e4 niist\u00e4.\nVakioleveill\u00e4 joukoilla on my\u00f6s monia sovelluskohteita. Niit\u00e4 on k\u00e4ytetty vanhanaikaisissa\nfilmiprojektoreissa filmin liikuttamiseen, Wankelin moottorissa kiertom\u00e4nt\u00e4n\u00e4\nsek\u00e4 poranteriss\u00e4. Mielenkiintoisin n\u00e4ist\u00e4 on poranter\u00e4, sill\u00e4 vakioleveisiin joukkoihin\npohjautuvilla terill\u00e4 on mahdollista porata neli\u00f6n mallisia reiki\u00e4. Vakioleveit\u00e4\njoukkoja voi l\u00f6yt\u00e4\u00e4 my\u00f6s Iso-Britannian kolikoista, joista 20 ja 50 pennyn kolikot ovat\nmuodoltaan vakioleveit\u00e4.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2019-06-10T07:11:11Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2019-06-10T07:11:11Z (GMT). No. of bitstreams: 0\n Previous issue date: 2019", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "43", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Vakioleve\u00e4t joukot", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201906103085", "language": "", "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": "", "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "joukot", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
|